Heine integral
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11—20 of 425 matching pages
11: 14.28 Sums
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§14.28(ii) Heine’s Formula
…12: Bibliography V
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An integral transform involving Heun functions and a related eigenvalue problem.
SIAM J. Math. Anal. 17 (3), pp. 688–703.
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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds.
Math. Comp. 23 (105), pp. 61–69.
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Modular hypergeometric residue sums of elliptic Selberg integrals.
Lett. Math. Phys. 58 (3), pp. 223–238.
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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Integral relations for Lamé functions.
SIAM J. Math. Anal. 13 (6), pp. 978–987.
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13: 17.6 Function
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Heine’s First Transformation
… ►Heine’s Second Tranformation
… ►Heine’s Third Transformation
… ►Heine’s Contiguous Relations
… ►§17.6(v) Integral Representations
…14: 28.18 Integrals and Integral Equations
§28.18 Integrals and Integral Equations
…15: 18.11 Relations to Other Functions
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§18.11(ii) Formulas of Mehler–Heine Type
…16: 19.35 Other Applications
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§19.35(i) Mathematical
►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute to high precision (Borwein and Borwein (1987, p. 26)). ►§19.35(ii) Physical
… ►17: 6.1 Special Notation
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►Unless otherwise noted, primes indicate derivatives with respect to the argument.
►The main functions treated in this chapter are the exponential integrals
, , and ; the logarithmic integral
; the sine integrals
and ; the cosine integrals
and .
18: 25.7 Integrals
§25.7 Integrals
►For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).19: 36.3 Visualizations of Canonical Integrals
§36.3 Visualizations of Canonical Integrals
►§36.3(i) Canonical Integrals: Modulus
… ►§36.3(ii) Canonical Integrals: Phase
►In Figure 36.3.13(a) points of confluence of phase contours are zeros of ; similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of ; similarly for other density plots in this subsection. …20: 6.14 Integrals
§6.14 Integrals
►§6.14(i) Laplace Transforms
… ►§6.14(ii) Other Integrals
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6.14.6
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►For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).